Lecture: Levered Firms

Lutz Kruschwitz & Andreas Loffler

Discounted Cash Flow, Section 3.2

Remark: The slightly expanded second edition (Springer, open access) has

different enumeration than the first (Wiley). We use Springer’s

enumeration in the slides and Wiley’s in the videos.,

Outline

3.2.1 Equity and debtCost of debtBook values

3.2.2 Earnings and taxesGross cash flowsCorporate income taxTax shield

3.2.3 Financing policies3.2.5 Default (insolvency)

Our assumptionsValuation of tax shieldCost of debt3.2.6 The finite example

Summary

,

Equity and debt 1

Let us have a closer look at levered firms: their value V lt equals

debt (Dt) plus equity (Et)

V lt = Dt + Et .

These are market values, not book values! Debt is granted attime t, after one period the debtor has to pay redemption andinterest It+1. Without default

It+1 = rf Dt .

(Default will be discussed later.)

3.2.1 Equity and debt, Cost of debt

Debt ratio and leverage ratio 2

Debt ratio and leverage ratio are two important numbers. Bothcan be uncertain:

debt ratio lt =Dt

V lt

leverage ratio Lt =Dt

Et

implies Lt =lt

1− lt.

3.2.1 Equity and debt, Cost of debt

Book values 3

We now introduce the book value of a company. These are thosevalues, with which the owners’ or creditors’ claims are to be foundin the balance sheets.

The book value Vl

t equals book value of debt (Dt) plus book valueof equity (E t)

Vl

t = Dt + E t .

Again debt ratio and leverage ratio can be defined

lt =Dt

Vl

t

and Lt =Dt

E t

.

3.2.1 Equity and debt, Book values

Earnings and taxes 4

= Earnings before taxes EBT

+ Interest rf D

= Earnings before interest and taxes EBIT

+ Accruals Accr

= Gross cash flow before taxes GCF

− Taxes Tax

− Investment expenses Inv

= Free cash flow FCF

Only the red items differ for levered and unlevered firms.

Assumption 3.2 (identical gross cash flows): There is nodifference between levered and unlevered firms with respect togross cash flows, accruals, investments.

3.2.2 Earnings and taxes, Gross cash flows

Corporate income tax 5

Earnings before taxes are the tax base. Hence we have for the taxpayments of the corporation

Tax = τ EBT . (3.8)

Because EBITu= EBIT

l, the tax payments of the levered and the

unlevered firm differ by

Taxu

t − Taxl

t = τ rf Dt−1.

This difference is called the tax shield from debt.

3.2.2 Earnings and taxes, Corporate income tax

Tax shield and DCF 6

Interest is determined by the riskless rate rf and debt Dt−1. Thetax shield is

τ rf Dt−1.

From our assumptions only debt can be uncertain in thismodel!=⇒ The level of tax shield is determined by the financing policy.

The aim of DCF is the valuation of these (uncertain) taxadvantages. Not less, but also not more.

3.2.2 Earnings and taxes, Tax shield

Financing policies 7

1. Autonomous financing: future amount of debt Dt is fixed.

2. Financing based on market values: evaluator sets thefuture debt ratios based on market values lt .

3. Financing based on book values: the future debt ratios tobook values lt are fixed.

4. Financing based on cash flows: amount of debt is based onthe firm’s cash flows.

5. Financing based on dividends: debt managed so thatpreviously determined dividend distributed.

6. Financing based on dynamical leverage ratio: evaluatorsets the future cash flow–debt ratios.

3.2.3 Financing policies,

Given debt policy 8

We do not want to answer the question as to which of thesefinancing policies is particularly close to reality. Further, we willnot discuss the question of which of the mentioned financingpolicies maximizes the value of the levered company (later we willsee: extended leverage increases company value).

Assumption 3.3 (given debt policy): The debt policy of the firm(although probably uncertain) is already prescribed.

3.2.3 Financing policies,

Value of tax shield: The cash flows 9Our aim is a general valuation equation for the tax shield.

Free cash flows differ only by tax payments (see slide 4). Using thedefinition of tax base (3.8)

FCFl

t = FCFu

t + τ rf Dt−1 (3.9)

From the fundamental theorem (Theorem 2.2) for levered andunlevered firms we get

V l ,ut =

EQ

[FCF

l,u

t+1 + V l,ut+1|Ft

]1 + rf

.

It follows that

V lt − V u

t =EQ

[τ rf Dt + V l

t+1 − V ut+1|Ft

]1 + rf

.

3.2.3 Financing policies,

Value of tax shield: Evaluation 10

Then

V lt = V u

t +EQ

[τ rf Dt |Ft

]1 + rf

+ . . .+EQ

[τ rf DT−1|Ft

](1 + rf )T−t

.

This gives finally

V lt = V u

t +τ rf EQ

[Dt |Ft

]1 + rf

+. . .+τ rf EQ

[DT−1|Ft

](1 + rf )T−t

. (3.11)

This is the basic equation for valuing the tax shield. It clearlyshows a dependence of the tax shield on the financing policy(future debt levels).

3.2.3 Financing policies,

Default 11

Default trigger can be, for example:

– lack of liquidity,

– debt greater than assets,

– expected lack of liquidity in the near future.

At the moment we will not specify the default trigger.

If bankruptcy occurs, there are three possibilities:

– restructuring the company,

– liquidation of the company, or

– sell-out of the remaining assets.

We will only assume that all conceivable developments were takeninto consideration when determining the company’s cash flows.

3.2.5 Default (insolvency),

What do we assume? 12

Owners have no personal liability.

Creditors and shareholders have identical information about thecompany, its estate, the firm’s cost of capital and its financingpolicy.

Assumption 3.5 (gross cash flows and default): The gross cashflows as well as the investment and accruals policy of the unleveredfirm do not differ from those of the firm in danger of default.

3.2.5 Default (insolvency), Our assumptions

Prioritization rules 13

Who gets what in case of default?

Assumption 3.6 (prioritization of debt): The tax office’s claimsrange before those of other creditors. The cash flows are alwayssufficient to at least pay off the tax debts in full.

Now a new notation is necessary:

Dt amount of credit outstanding in t

Rt+1 amount that is paid back in t + 1

It+1 interest paid in t + 1

3.2.5 Default (insolvency), Our assumptions

Taxes and default 14

The tax office allows interest It+1 to be deducted from the taxbase. On the other hand, the cancellation of debt

Dt − Dt+1︸ ︷︷ ︸should be paid back

− Rt+1︸︷︷︸was paid back

adds to the tax base (‘recapitalization gain’). Hence, the tax duefor a firm in danger of default is

Taxl

t+1 = τ(GCF t+1 − Accrt+1 − It+1︸︷︷︸

interest,deducted

+ Dt − Dt+1 − Rt+1︸ ︷︷ ︸gain,added

).

3.2.5 Default (insolvency), Valuation of tax shield

Value of the tax shield and default 15

Again from slide 4 we have

FCFl

t+1 = FCFu

t+1 + Taxu

t+1 − Taxl

t+1

= FCFu

t+1 + τ(It+1 + Rt+1 + Dt+1 − Dt

)and the main valuation equation now reads

V lt = V u

t +T∑

s=t+1

τ EQ

[Is + Rs + Ds − Ds−1|Ft

](1 + rf )s−t

. (3.15)

This does not look like (3.11)?! But let us see. . .

3.2.5 Default (insolvency), Valuation of tax shield

Fundamental theorem for the creditors 16

The debtholders behave rational. Therefore, at time s

Ds−1 =EQ [value in s|Fs−1]

1 + rf=

EQ

[Rs + Ds + Is |Fs−1

]1 + rf

and from rule 2 and rule 5 for all t ≤ s

rf EQ

[Ds−1|Ft

]= EQ

[Is + Rs + Ds − Ds−1|Ft

].

And the rhs is the nominator from the main valuation equation(3.15) above!

3.2.5 Default (insolvency), Valuation of tax shield

Fundamental theorem for the creditors 17

Even in the case of default the valuation equation (3.11)

V lt = V u

t +τ rf EQ

[Dt |Ft

]1 + rf

+ . . .+τ rf EQ

[DT−1|Ft

](1 + rf )T−t

holds.

Or: default does not make the DCF theory fail. The difficulties oftaking default into consideration lie much more in the fact that therelevant financing policies must be formulated with care.

3.2.5 Default (insolvency), Valuation of tax shield

Remark: Cost of debt 18

Someone who invests Dt today is entitled to payments amountingto Dt + It+1 less remission of debts Dt − Dt+1 − Rt+1. Hence,

Definition 3.4 (cost of debt): The cost of debt of a levered firmis

kDt =E[Dt+1 + It+1 + Rt+1|Ft ]

Dt

− 1.

If there is no default kDt = rf .

We do not require the cost of debt to be deterministic today. Costof debt will not be used itself to determine the value of firms.

3.2.5 Default (insolvency), Cost of debt

Finite example 19

Provisional leverage policy may be

D0 = 100, D1 = 100, D2 = 50.

Tax rate is τ = 50%.

Bankruptcy enters in if in state ω,{FCF

l

t(ω) < It(ω) + Dt−1 − Dt for a levered firm,

FCFu

t (ω) < 0 for an unlevered firm.

Here: bankruptcy is equal to lack of liquidity.

3.2.5 Default (insolvency), 3.2.6 The finite example

Shareholder’s claims 20Does bankruptcy occur? FCF

u

t (ω) + τ rfDt−1 − (1 + rf )Dt−1 +Dt

105

85

77

55

33

141.1

44.3

92.7

– 4.1

3.2.5 Default (insolvency), 3.2.6 The finite example

The state dd 21

In all states except dd (and following) the creditors demand

ω = dd =⇒ kD,nominalt (ω) = rf .

Only in state ω = dd bankruptcy can occur. Hence,

kD,nominalt (dd) > rf .

We now want to determine kD,nominalt (dd).

3.2.5 Default (insolvency), 3.2.6 The finite example

Determining kD,nominalt (dd) 22

Let us look at the states that follow dd . At state ddd the claimscannot be paid off in full. The cash flow is necessary to payinterest and pay back the loan. Therefore,

FCFl

3(ddd) + D3︸︷︷︸=0

= R3(ddd) + I3(ddd).

On the other hand, since the tax office claims have priority,

FCFl

3(ddd) = FCFu

3(ddd) + τ(I3(ddd) + R3(ddd) + D3︸︷︷︸=0

−D2)

which finally gives

FCFl

3(ddd) =1

1− τ

(FCF

u

3(ddd)− τD2

)= 46.8.

3.2.5 Default (insolvency), 3.2.6 The finite example

Determining kD,nominalt (dd) 23

At state ddu the creditors get

(1 + kD,nominal2 (dd))D2 = R3(ddu) + I3(ddu).

Using the fundamental theorem we must have

D2 =EQ [R3 + I3|F2]

1 + rf

which finally gives

kD,nominal2 (dd) ≈ 32.962%

The resulting cost of debt are

kD2 (dd) =

(1+kD,nom

2 (dd))D2 P3(u|dd)+ 1

1−τ

(FCF

u

3 (ddd)−τD2

)P3(d |dd)

D2− 1

≈(1+0.32962)×50×0.5+ 1

1−0.5 (48.4−0.5×50)×0.5

50−1≈13.281%.

3.2.5 Default (insolvency), 3.2.6 The finite example

Resulting cash flows 24

From these results we get

FCFl

3(ddu) = FCFu

3(ddu) + τ(I3(ddu) + R3(ddu) + D3 − D2)

= 145.2 + 0.5 · 0.32962 · 50≈ 153.44 .

This is enough to pay the (nominal) interest rate of almost 33% inddu.

Notice, that the binomial tree is now not recombining.

3.2.5 Default (insolvency), 3.2.6 The finite example

Cash flows FCFlwith default risk 25

115

95

137

115

93

196.1

99.3

147.7153.4

46.8

3.2.5 Default (insolvency), 3.2.6 The finite example

Summary 26

We consider levered and unlevered firms with identical gross cashflows.

The tax advantage for the levered firm (no default) is τ rf Dt−1.

The financing policy of the firm determines the value of the taxshield.

DCF remains valid even in the case of default. But then therelevant financing policy must be handled with care.

Summary,